Saturation and connection would likely help the most, and I know that's a pretty obvious observation. Everton is not a household name in the US, where only the top world clubs are. For people looking for a club to follow, a lot will want to remedy the lack of authenticity in simply "picking" one by not picking a top 4 club. So that puts Everton in a decent window of clubs that are competitive but with more to achieve. Many adopters will want to feel they're on the ride somewhere, not picking a team already on top.
In the near term, utilizing the US supporter clubs to evangelize for the club is a good concept. Here in Atlanta, one of the two biggest soccer/football bars in the city is the home of the Everton club. Atlanta, likely in part due to that reason, has one of the bigger supporter clubs in the US. You can't go to that bar to watch on a Sat/Sun without seeing Everton stuff hanging from railings. It's a natural way to find a club to support, at least in this context.
Another tour would help, and picking out soft spots of undeveloped support would be wise. Colleges, growing cities in the Southeast and Texas where soccer/football is only now becoming more popular, etc. Americans looking for a club to be a part of are looking for a way to make that connection genuine. Howard gives one, local clubs provide another. Partnership with MLS would help, but a lot of those people are aligned already. Honestly, even a casual relationship or interest by Everton with other American sports teams on any level (pro, college, baseball, American football,etc.) may be more productive ("Seamus Coleman tweets support for Ohio State? Looks like Everton is the club for me" says everyone from Ohio).
Not sure how to make that happen in a systematic way, or how to avoid alienation, but I think that is the type of connection that will provide returns.